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Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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Double dual Spaces and the annihilator

I am revising linear algebra, and am a bit stuck on this problem. So, if you define a natural isomorphism $f$ between $V$ and its double dual $V''$, you get $$f:V\rightarrow V''$$ $$v\mapsto E_v$$ ...
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Adjoint of a linear isomorphic functional is an isomorphism

I have this exercise to solve: Let $E$ and $F$ be normed spaces and let $T \in L(E,F)$, where $L(E,F)$ denotes the set of all bounded linear operators from $E$ to $F$. $T^*$ is the dual operator of ...
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The dual space of $C(Y,\mathbb R)$ when $Y$ is a complete and separable metric space

Just to be confirmed what is the dual space of $C(Y,\mathbb R)$ i.e the vector space of all continuous functions $f: Y\to \mathbb R$ when $Y$ is complete and separable metric space? Is it the same ...
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Linear algebra-question about the annihilator for a dual space

I am revising linear algebra and am stuck on a question. I want to show that if $v \notin U$ then there is some $f \in U^0$ such that $f(v) \neq 0$. I am confused about how to go about proving this, ...
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Hamel dimension of a vector space, and dimension of the dual

I have the following (possibly trivial) observation: Let $K$ be an $\mathbb{F}$-vector space (I believe the argument also works for free modules), and let $X\subseteq K$ be it's basis with ...
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find the dual basis of $\mathbb R_{\leq2}[x]$ given a basis $B=(x^2-1,x^2-x,x^2+x)$

find the dual basis of $\mathbb R_{\leq2}[x]$ given a basis $B=(x^2-1,x^2-x,x^2+x)$ I know what a dual basis means, however I seem to miss something. here's my attempt: I need to find a basis $B^\...
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Intersecting the annihilator of a vector space with the first orthant

Let $V \subset \mathbb{R}^n$ be a subvector space, and $V^{\perp}$ be its annihilator in the dual $\mathbb{R}^{n*}$ of $\mathbb{R}^n$. I need to show the following implication: $$ V^{\perp} \cap \...
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transition matrix of dual basis

Q :Let $\left\lbrace v_i \right\rbrace^n_{i=1}$ and $\left\lbrace w_i \right\rbrace^n_{i=1}$ be basis of V and also let $\left\lbrace \phi_i \right\rbrace^n_{i=1}$ and $\left\lbrace \sigma_i \right\...
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Trying to understand some formulas about dualities, specifically $\operatorname{Hom}_{\Lambda}(Y,\nu X) \cong D\operatorname{Hom}_{\Lambda}(X,Y)$.

I'm reading $\tau$-tilting by Adachi, Iyama and Reiten. There is a particular isomorphism of hom-sets which I'm having trouble understanding. The setup: $\Lambda$ is a finite-dimensional $k$-...
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related Theorem of linear functional

Q :Let V be a finite-dimensional vector space over a field k and let $V^\ast=Hom(V,k)$ be the dual space of V. Let $\left\lbrace v^i \right\rbrace^n_{i=1} $ be the dual basis of $V^\ast$. Then prove ...
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Annihilator of the Kernel is equal to image of the dual map

I'm revising Linear Algebra and am stuck on this question. Supposing that $T:V \rightarrow W$ is a linear map, and that V is finite dimensional, I want to prove that $Im(T')=(Ker(T))^0$. I know ...
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Show that if the 0-functional is the only linear bounded functional that is 0 on a set, the set is dense

I want to show that $$(\forall f\in X^*: f|_F \equiv 0 \Rightarrow f\equiv 0) \Leftrightarrow F \text{ dense in }X.$$ I have proven the direction $\Leftarrow$. But I have problems with the other ...
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How to show dual norm of subgradient of a $L$-Lipschitz convex function is bounded by $L$?

I am studying the monograph Online Learning and Online Convex Optimization. At page 133, the author has the following Lemma: $\textbf{Lemma 2.6.}$ Let $f: S \rightarrow \mathbb{R}$ be a convex ...
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In a Topological Vector Space T0 implies T3½ (completely regular)? And other separation properties.

I will describe my doubt. I know that in a TVS T1 implies T2. Now since a TVS admits a uniformisable topology, we have that T2 implies the uniform structure is separating. Now a separating uniform ...
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Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
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Show that $B(X,Y^*)$ and $B(Y,X^*)$ are isometrically isomorphic.

If $X$ and $Y$ are normed spaces then we define. $$B(X,Y)= \{ f:X\rightarrow Y | f :\text{ f is a linear operator and bounded }\}$$ $X^*= \{f:X \rightarrow \mathbb{R} | \text{ f is a linear ...
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For a subspace $U$ of $V$, is the annihilator of the annihilator of $U$ equal to the image of $U$ under the canonical isomorphism $V \to V^{**}$?

So, with the annihilator of a set $U$ defined as Anni($U$) = $\{f \in V^* \: | \: \text{for all $u \in U$: }f(u) = 0\}$, and the canonical isomorphism $\varphi: V \to V^{**}$ given by $\varphi(v)(w^*) ...
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Are these two norms on the dual space of a Hilbert space equivalent?

Let $\mathcal{H}$ be a Hilbert space, and $\mathcal{H}^*$ its topological dual space (the space of continuous linear forms on $\mathcal{H}$). The exists a conjugate-linear isometry between these two ...
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understanding Hilbert Sobolev spaces

We denote the space $\dot{H}^k$ for the homogeneous Sobolev space and $H^k$ for the inhomogeneous Sobolev space where $H^k=W^{k,2}$. It is true that de dual of $\dot{H}^k$ can be identified with $H^k$...
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An isomorphism between the dual space of a Hilbert space and a larger space containing the same Hilbert space

Let $G \subset H \subset F$ be three Hilbert spaces such that the smaller ones are continuously and densely embedded into the larger ones. Furthermore, assume $$|\langle h, g\rangle_H|\le \|h\|_{F}\|g\...
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Hahn-Banach Theorems Applications

Please, if anyone can help me with some useful tips to solve this aim: Let $K^1,...,K^n$ closed convex sets containing the origin of a normed space $E$, and let $c_1,...c_n$ positive real numbers. ...
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Under what conditions does $L^{1}(X)$ have a predual?

I know this question has been asked a million times—but they seem to always be with some special flair. I've looked at many and cannot extract from them an answer to my plain question: Question: Let $...
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Describe the Bidual of Real Polynomials

Trying to understand (bi)duals of infinite-dimension vector spaces, I stumbled over the very concrete example of $\mathbb{R}[X]$, the (formal) polynomials over $\mathbb{R}$ or, equivalently, the space ...
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Connection between a representation and its isomorphic dual

Suppose we have a complex irreducible representation $(V, \phi)$ of a finite group $G$. (Where $\phi : G \rightarrow \text{Aut}_{\mathbb{C}}(V)$). Suppose $V$ it is isomorphic to its dual $V^*$, and ...
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Quadratic Forms and Dual Space

Suppose we have $V$, a finite $k$-vector space, where k is a field with $char(k) \neq 2$. And we have a quadratic form $Q$ on $V$. If $Q$ is non-degenerate then the map $q_v:V \to V^*$ sending $v \in ...
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Linear algebra - find dual space base

I'm trying to solve what seems to be a simple problem, but I cannot find the right way to approach it. Here it is: Let $V$ be a vector space of all polynomials of degree $0$ or $1$. We define the ...
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The definition and norm of $(\mathbb{R}^*)^2$

Let $X(x,y)\in E = \Bbb R^2$ with the norm $||X||_1 = |x | + |y |$ Find $E^*$ and its norm, where $E^*$ is the topological dual, and compute $F((1,0))$ where $F(u) =\{L\in E^*, ||L||_{E^∗} = ||u||_E, ...
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Representation of the elements of the dual space of the product of topological vector spaces

Assume $(X_i,\mathcal{T}_i)$, $i \in I$ is a family of topological vector spaces and $X:=\prod\limits_{i\in I} X_i$ with the product topology $\mathcal{T}$. Let $\pi_i: X \rightarrow X_i$ be the ...
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convex subset of topological dual

Let $(E, ||.||_E )$ be a banach space, and $E^∗$ its topological dual. For $u ∈ E$, prove that $F(u) =\{L\in E^*, ||L||_{E^∗} = ||u||_E, \left<L, u\right> = ||u||^2_E \}$ is convex. Let $t\in ...
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Questions about deriving the dual space of $l^{1}$

I am an engineering student and I am reading the book "Introductory Functional Analysis " by kreyszig and am lost in the proof of finding the dual space of the $l^{1}$ space . Here is how author ...
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A covariant functor sending every finite-dimensional vector space to its dual?

Wikipedia defines a particular map between two particular objects as an unnatural isomorphism if it is an isomorphism that cannot be extended to a natural transformation on the entire category. Dual ...
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Finding the basis for the annihilator

I am revising Linear Algebra, specifically dual bases, and am stuck on this part of the question. You are given that V is a real vector space, and U is a subspace of V. You are then told that {$x_1$...
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Confusion about Hahn-Banach

I am confused about something related to Hahn-Banach. According to my book, one corollary of H-B is that for $X$ a real or complex normed space, there exists $f \in X'$ such that $\|f\| = 1$ and $f(x) ...
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Finite Rank Operator in Normed Space, not necessarily Hilbert neither Banach

Suppose that $E$ and $F$ are normed spaces and $T:E \rightarrow F$ is a bounded linear operator. I NEED TO SHOW WHAT FOLLOWS: If there are $n\in \mathbb{N}, f_{1}, ..., f_{n}\in E^{\ast}$ (dual of $...
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How to show that $\mathcal{F} \otimes \mathcal{F}^\vee \cong \mathcal{O}_X$

Let $\mathcal{F}$ be a rank $1$ locally free sheaf. If we define $\mathcal{F}^\vee = Hom_{\mathcal{O}_X}(F, \mathcal{O}_X)$, then how would one go about showing that $\mathcal{F} \otimes \mathcal{F}^\...
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Prove that the dual basis is linearly independent

Q: Suppose $\{e_1,\ldots,e_n\}$ is a basis of a vector space V over the field F . Let $e^1,\ldots,e^n\in$ $V^*$ be the linear functionals defined by $$ e^i(e_j) = \begin{cases} c, i=j \\ 0, i\neq j \...
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A question about the dual of super vector space

Let $V$ be a vector space over a field $K$. Denote the dual of $V$ by $V^{*}$, that is $V^*=Hom_K(V,K)$. Suppose there is a morphism $\alpha: V \rightarrow V$. Then we know $\alpha$ induces a morphism ...
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$X$ Banach, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$ implie that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.

Exercise : Let $X$ be a Banach space, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$. Show that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$. Attempt-Discussion : I know that a sequence $...
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$E=L^1(\Bbb R) \Rightarrow E'=L^{\infty}(\Bbb R) \Rightarrow E''\ne L^1(\Bbb R) $

As an example of a banach space that is not reflexive we have $E=L^1(\Bbb R)$ As a proof I had this: $E=L^1(\Bbb R) \Rightarrow E'=L^{\infty}(\Bbb R) \Rightarrow E''\ne L^1(\Bbb R)$ I don't ...
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Bilinear Maps and their relationships with dual bases

I have a theorem without proof. I have searched many books and tried on myself, but i still dont have the solution. Let M and N F-vector spaces, T be a base of M ,S be a base of N such that dimension ...
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Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra

So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan. The problem is the following: Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
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1answer
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If operator is closed and densely defined then $D(A^*)^\perp = \{0\}$

I'm a bit rusty in my Functional Analysis and couldn't solve this question: Let $X$ be a Banach space (over either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Show that, if $A:D(A) \...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
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Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
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Finding a basis of the annihilator of a subspace

Let $V\subset \mathbb{R}^4$ be the subspace spanned by $e_1+e_2+e_3+e_4$ and $e_1+2e_2+3e_3+4e_4$. Find a basis of the subspace $V^{\circ}$ in the dual space $(\mathbb{R})^*$. My attempt: Let $a = ...
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Why are $T^k(V)$ and $V^* \otimes… \otimes V^*$ just isomorphic?

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* \otimes... \otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $\epsilon^1,.....
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Commutative diagrams for vector spaces, dual spaces, and adjoint of linear maps

I'm looking for a commutative diagram explaining the relationship between a vector space, $X$ the dual space, $X^*$, the double dual, $X^{**}$. I'm also looking for a diagram explaining the ...
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Value of linear function $\alpha(h)$ defined in dual basis

Let $V = \mathbb{R}[x]_{\leq 2}$ and $\{\varepsilon_1, \varepsilon_2, \varepsilon_3\}$ - dual basis in $V^*$ of basis $\{-1 - x -2x^2, 7 + 6 x + 13x^2, 9 + 15x + 23x^2\}$ in $V$. Also $\{f_1, f_2, ...
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Bra’s, Ket’s, a Hilbert Space and it’s Dual.

So I’m trying to get this all straightened out in my head. In Quantum mechanics we use a Hilbert Space $\mathcal{H}$ as our vector space and we say that its elements is the set of Ket’s $\left|\psi\...
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Construct a natural map $l_1 → l_{\infty}^∗$

I know from Show that $(l_1)^* \cong l_{\infty}$ that $l^∗_1$ is isometrically isomorphic to $l_{\infty}$. Indeed we can show that a map $L: l_{\infty} \rightarrow (l_1)^*$ given by $$L(x)(y) = \sum_{...